Abstract Algebra MCQs

Q1: Which of the following is a group under addition?

(A) ℕ
(B) ℤ
(C) ℝ⁺
(D) ℚ⁺

Answer: (B) ℤ

Q2: Which of the following is a field?

(A) ℤ
(B) ℝ
(C) ℕ
(D) ℚ

Answer: (D) ℚ

Q3: Which of the following is an abelian group?

(A) The set of integers under multiplication
(B) The set of non-zero real numbers under multiplication
(C) The set of 2×2 invertible matrices under matrix multiplication
(D) The set of integers under addition

Answer: (D) The set of integers under addition

Q4: If a group has an identity element e, which of the following is true for any element a in the group?

(A) a * e = a
(B) a * e = e
(C) a * e = e * a
(D) a * a = e

Answer: (A) a * e = a

Q5: Which of the following sets is not a subgroup of ℝ under addition?

(A) ℤ
(B) ℝ
(C) ℝ⁺
(D) ℚ

Answer: (C) ℝ⁺

Q6: What is the order of the group ℤ₆ under addition modulo 6?

(A) 6
(B) 3
(C) 2
(D) 1

Answer: (A) 6

Q7: A ring is a set R with two binary operations. Which of the following is a necessary condition for R to be a ring?

(A) R must have an identity element for multiplication.
(B) R must be abelian under addition.
(C) R must be commutative under multiplication.
(D) R must have an identity element for addition.

Answer: (B) R must be abelian under addition.

Q8: Which of the following is an example of a non-abelian group?

(A) ℤ
(B) ℝ⁺
(C) The set of 2×2 invertible matrices under matrix multiplication
(D) The set of real numbers under addition

Answer: (C) The set of 2×2 invertible matrices under matrix multiplication

Q9: If a ring has a multiplicative identity, then it is called:

(A) A field
(B) A division ring
(C) A commutative ring
(D) A unital ring

Answer: (D) A unital ring

Q10: Which of the following is true about a cyclic group?

(A) It contains only one element
(B) It is isomorphic to ℤ
(C) It is always finite
(D) All of the above

Answer: (B) It is isomorphic to ℤ